(2^n+1)/3

[Gottfried Helms]

Am 20.05.05 08:50 schrieb kohmoto:

> 
>     Hello, Seqfans
>     Once I realized that if M_n is a Mersenne prime then (M_n+2)/3 is
> also prime. 2<n
>     And I knew that it is called "Bateman and Shefridge and Wagstaff's
> conjecture ".
>     Does anyone know the exact description of it?
>  
An addition to my previous post:

You can rewrite the 3rd part of the "bateman-..."-conjecture, which
states
   3)  (2^p+1)/3 = q    is prime

into
    2^p + 1  = 3*q

and the cyclic length of 2^p+1 seems always to be

    L2(2^p+1) = 2*p

so that

    2*p = l2(3*q)

which, for the multiplicity in my assumption, means

    2*p = lcm(l2(3),l2(q)) = lcm(2,l2(q))

This is immediately true for a mersenne-prime q, where
  q = 2^p-1

since then
  L2(q) = L2(2^p-1) = p

and then

   2*p = lcm(2,p) = 2*p

Possibly using the 1st condition of the "Bateman..."-conjecture
one can narrow this down to a *required* condition by just algebraic
manipulation of the above formulas - always on the basis of
the -assumed- uniqueness cycle-length of mersenne-primes.

I'll try with this a bit today.


Gottfried Helms